AITAH for making a dairy free cake with eggs?

My friend invited me over for dinner. Our tradition is to bring food when invited to dinner. I asked if I should bring a dessert. She said that would be great but it needs to be dairy free because her daughter stopped eating dairy.

I know a great cake recipe with no dairy. I brought it to the dinner and it was a big hit. My friend asked for the recipe and I told her. When I got to the eggs her daughter became upset and ran upstairs.

My friend asked why I put eggs in the cake. I said the eggs were responsible for the spongy texture. She said the cake wasn’t dairy free. I said it was. There is no milk or milk products in the cake, no cheese, no butter, etc…

My friend said eggs are dairy. I was confused. Dairy is milk and milk products to my understanding. She said dairy is anything that comes from an animal. I asked if honey is dairy then. She said no, because bees are insects. I apologized, but I could tell she was still upset.

Am I the asshole? I’ve never heard of eggs being dairy before.